Motion Under Constant Power

Question

A machine delivers a constant power of 16 W to a 2 kg object that starts from rest on a frictionless surface. What speed does the object attain after 4 s of operation?

RankGuru · Accepted Answer

When power is held constant, the energy delivered grows linearly with time, $W = Pt$, and by the work-energy theorem this equals the kinetic energy gained from rest, $Pt = 	frac{1}{2}mv^2$. Solving for speed gives $v = \sqrt{2Pt/m}$, which shows that under constant power the speed rises in proportion to the square root of time, not linearly. Substituting $P = 16	ext{ W}$, $t = 4	ext{ s}$, and $m = 2	ext{ kg}$ gives $v = \sqrt{(2 	imes 16 	imes 4)/2} = \sqrt{64} = 8	ext{ m/s}$. The option 16 m/s reports $v^2$ without the square root. The option 4 m/s wrongly assumes speed grows linearly with time. The option 32 m/s omits the division by mass. The square-root dependence on time carries clear physical meaning: as the object speeds up, the fixed power must be shared with an ever-larger velocity, so the driving force and hence the acceleration steadily diminish. This contrasts sharply with constant-force motion, where the speed grows linearly with time and the acceleration remains fixed throughout. This is the NCERT result for one-dimensional motion under constant power. As a check, the energy delivered is $16 	imes 4 = 64	ext{ J}$, matching $	frac{1}{2}(2)(8)^2 = 64	ext{ J}$ exactly.

Motion Under Constant Power

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